Cool stuff: 2^n - 3^k and its divisors are showing up (with n length of the parity vector and k number of 1s) @oros @mathkook
Below, find all the kernels for parity vector [0,0,0,0,1,1], for each valued path in the kernel I’ve expressed the function computed by the valued path in terms of their slope (always 3^k/2^n) and fix point (i.e. \frac{\text{intercept}}{1-\text{slope}}).
It’s quite beautiful because the denominator of the functions’ fixpoints are the divisors of 55, i.e. 55, 11, 5, 1.
Then you can see how in each kernel with same denominator there is just a different of -1 in each numerator (e.g. -16/55 in Kernel 1, -17/55 in Kernel 2, -18/55 in Kernel 3, -19/55 in Kernel 4). Here in Kernel 5 you get all -1/11 to -10/11, which is cool.
As per the conjecture, these kernels are the same for all parity vectors with same number of 1s and in this case it turns out that the parity vector (i.e. (('←', 0), ('←', 0), ('←', 0), ('←', 0), ('↙', 4), ('↙', 4)) is in the component) of Kernel 5.
Hence I think that the structure of kernels must be easy to describe, in terms of divisors of 2^n - 3^k (hopefully the size of each kernel, the number of kernels of each size and the numerators are also easy to predict).
Kernel 0, size 1
(('←', 0), ('←', 0), ('←', 0), ('←', 0), ('↙', 0), ('↙', 0)) (9/64, 0)
Kernel 1, size 10
(('←', 0), ('←', 0), ('←', 0), ('←', 0), ('↙', 3), ('↙', 4)) (9/64, -16/55)
(('←', 0), ('←', 0), ('←', 1), ('←', 0), ('↙', 0), ('↙', 1)) (9/64, -36/55)
(('←', 0), ('←', 1), ('←', 0), ('←', 1), ('↙', 1), ('↙', 2)) (9/64, -26/55)
(('←', 0), ('←', 1), ('←', 1), ('←', 0), ('↙', 4), ('↙', 0)) (9/64, -6/55)
(('←', 0), ('←', 1), ('←', 1), ('←', 1), ('↙', 5), ('↙', 5)) (9/64, -46/55)
(('←', 1), ('←', 0), ('←', 0), ('←', 0), ('↙', 0), ('↙', 3)) (9/64, -41/55)
(('←', 1), ('←', 0), ('←', 0), ('←', 1), ('↙', 4), ('↙', 4)) (9/64, -1/55)
(('←', 1), ('←', 0), ('←', 1), ('←', 1), ('↙', 2), ('↙', 0)) (9/64, -21/55)
(('←', 1), ('←', 1), ('←', 0), ('←', 1), ('↙', 5), ('↙', 1)) (9/64, -51/55)
(('←', 1), ('←', 1), ('←', 1), ('←', 0), ('↙', 1), ('↙', 5)) (9/64, -31/55)
Kernel 2, size 10
(('←', 0), ('←', 0), ('←', 0), ('←', 0), ('↙', 0), ('↙', 3)) (9/64, -32/55)
(('←', 0), ('←', 0), ('←', 1), ('←', 0), ('↙', 3), ('↙', 5)) (9/64, -52/55)
(('←', 0), ('←', 0), ('←', 1), ('←', 1), ('↙', 2), ('↙', 0)) (9/64, -12/55)
(('←', 0), ('←', 1), ('←', 0), ('←', 0), ('↙', 3), ('↙', 2)) (9/64, -2/55)
(('←', 0), ('←', 1), ('←', 0), ('←', 1), ('↙', 5), ('↙', 1)) (9/64, -42/55)
(('←', 1), ('←', 0), ('←', 0), ('←', 1), ('↙', 1), ('↙', 2)) (9/64, -17/55)
(('←', 1), ('←', 0), ('←', 1), ('←', 1), ('↙', 5), ('↙', 5)) (9/64, -37/55)
(('←', 1), ('←', 1), ('←', 0), ('←', 0), ('↙', 0), ('↙', 0)) (9/64, -27/55)
(('←', 1), ('←', 1), ('←', 1), ('←', 0), ('↙', 4), ('↙', 3)) (9/64, -47/55)
(('←', 1), ('←', 1), ('←', 1), ('←', 1), ('↙', 2), ('↙', 4)) (9/64, -7/55)
Kernel 3, size 10
(('←', 0), ('←', 0), ('←', 0), ('←', 0), ('↙', 3), ('↙', 1)) (9/64, -48/55)
(('←', 0), ('←', 0), ('←', 0), ('←', 1), ('↙', 1), ('↙', 2)) (9/64, -8/55)
(('←', 0), ('←', 0), ('←', 1), ('←', 1), ('↙', 5), ('↙', 5)) (9/64, -28/55)
(('←', 0), ('←', 1), ('←', 0), ('←', 0), ('↙', 0), ('↙', 0)) (9/64, -18/55)
(('←', 0), ('←', 1), ('←', 1), ('←', 0), ('↙', 4), ('↙', 3)) (9/64, -38/55)
(('←', 1), ('←', 0), ('←', 1), ('←', 0), ('↙', 0), ('↙', 4)) (9/64, -13/55)
(('←', 1), ('←', 0), ('←', 1), ('←', 1), ('↙', 2), ('↙', 3)) (9/64, -53/55)
(('←', 1), ('←', 1), ('←', 0), ('←', 0), ('↙', 3), ('↙', 5)) (9/64, -43/55)
(('←', 1), ('←', 1), ('←', 0), ('←', 1), ('↙', 2), ('↙', 0)) (9/64, -3/55)
(('←', 1), ('←', 1), ('←', 1), ('←', 1), ('↙', 5), ('↙', 2)) (9/64, -23/55)
Kernel 4, size 10
(('←', 0), ('←', 0), ('←', 0), ('←', 1), ('↙', 4), ('↙', 0)) (9/64, -24/55)
(('←', 0), ('←', 0), ('←', 1), ('←', 0), ('↙', 0), ('↙', 4)) (9/64, -4/55)
(('←', 0), ('←', 1), ('←', 0), ('←', 0), ('↙', 3), ('↙', 5)) (9/64, -34/55)
(('←', 0), ('←', 1), ('←', 1), ('←', 0), ('↙', 1), ('↙', 1)) (9/64, -54/55)
(('←', 0), ('←', 1), ('←', 1), ('←', 1), ('↙', 5), ('↙', 2)) (9/64, -14/55)
(('←', 1), ('←', 0), ('←', 0), ('←', 0), ('↙', 0), ('↙', 0)) (9/64, -9/55)
(('←', 1), ('←', 0), ('←', 0), ('←', 1), ('↙', 1), ('↙', 5)) (9/64, -49/55)
(('←', 1), ('←', 0), ('←', 1), ('←', 0), ('↙', 4), ('↙', 3)) (9/64, -29/55)
(('←', 1), ('←', 1), ('←', 0), ('←', 1), ('↙', 5), ('↙', 4)) (9/64, -19/55)
(('←', 1), ('←', 1), ('←', 1), ('←', 1), ('↙', 2), ('↙', 1)) (9/64, -39/55)
Kernel 5, size 10
(('←', 0), ('←', 0), ('←', 0), ('←', 1), ('↙', 1), ('↙', 5)) (9/64, -8/11)
(('←', 0), ('←', 0), ('←', 1), ('←', 0), ('↙', 3), ('↙', 2)) (9/64, -4/11)
(('←', 0), ('←', 1), ('←', 0), ('←', 0), ('↙', 0), ('↙', 3)) (9/64, -10/11)
(('←', 0), ('←', 1), ('←', 0), ('←', 1), ('↙', 4), ('↙', 4)) (9/64, -2/11)
(('←', 0), ('←', 1), ('←', 1), ('←', 1), ('↙', 2), ('↙', 1)) (9/64, -6/11)
(('←', 1), ('←', 0), ('←', 0), ('←', 0), ('↙', 3), ('↙', 4)) (9/64, -5/11)
(('←', 1), ('←', 0), ('←', 1), ('←', 0), ('↙', 1), ('↙', 1)) (9/64, -9/11)
(('←', 1), ('←', 0), ('←', 1), ('←', 1), ('↙', 5), ('↙', 2)) (9/64, -1/11)
(('←', 1), ('←', 1), ('←', 0), ('←', 1), ('↙', 2), ('↙', 3)) (9/64, -7/11)
(('←', 1), ('←', 1), ('←', 1), ('←', 0), ('↙', 4), ('↙', 0)) (9/64, -3/11)
Kernel 6, size 1
(('←', 0), ('←', 1), ('←', 1), ('←', 0), ('↙', 1), ('↙', 4)) (9/64, -2/5)
Kernel 7, size 1
(('←', 1), ('←', 0), ('←', 0), ('←', 1), ('↙', 4), ('↙', 1)) (9/64, -3/5)
Kernel 8, size 1
(('←', 1), ('←', 1), ('←', 0), ('←', 0), ('↙', 3), ('↙', 2)) (9/64, -1/5)
Kernel 9, size 1
(('←', 0), ('←', 0), ('←', 1), ('←', 1), ('↙', 2), ('↙', 3)) (9/64, -4/5)
Kernel 10, size 1
(('←', 1), ('←', 1), ('←', 1), ('←', 1), ('↙', 5), ('↙', 5)) (9/64, -1)